STUDIES ON IMPRECISE ECONOMIC ORDER QUAN

Volume 4, No. 8, August 2017
Journal of Global Research in Mathematical Archives
UGC Approved Journal
RESEARCH PAPER
Available online at http://www.jgrma.info

STUDIES ON IMPRECISE ECONOMIC ORDER QUANTITY MODEL USING
INTERVAL PARAMETER
Asim Kumar Das1*,Tapan Kumar Roy2.
* Email id: asd.math@gmail.com
Department of Applied Mathematics, Indian Institute of Engineering Science and Technology, Shibpur, Howrah,
West-Bengal, India, 711103.
Abstract: In this paper, we introduce an imprecise economic order quantity (EOQ) model with demand, holding cost and set up cost are assumed
as an interval number. We consider the parameters of the proposed model with imprecise data as form of interval number. The proposed EOQ
model is presented with impreciseness of parameters by introducing parametric functional form of interval number and then solves the problem
by geometric programming technique. Numerical example is presented to support of the proposed approach.
Keywords: EOQ model, holding cost, set up cost, Interval number, Geometric programming

INTRODUCTION
An inventory deals with decision that minimize the cost function or maximize the profit function. For this purpose the task is to
construct a suitable mathematical model of the real life Inventory system, such a mathematical model is based on various

assumption and approximation. This type of imprecise data is not always well represented by random variables selected from
probability distribution. So decision making methods under uncertainty are needed. To deal with this uncertainty and imprecise
data, the concept of fuzziness can be applied.
In ordinary inventory model it considers all cost parameter like set-up cost, carrying cost (holding cost), shortages etc are as fixed.
But in real life situation it will have some fluctuations. So far most of the researchers consider the EOQ model in precise
environment but in reality, data cannot be recorded or collected precisely due to human errors or some unexpected situations. In
real life, it is not always possible to obtain the precise information about inventory parameters. The inventory cost parameters
such as holding cost, set up cost, production cost, and reworking cost are rather assumed to be flexible in realistic sense. It is quite
natural that the parameters in the EOQ model are occurred in imprecise way. This type of imprecise data is not always well
represented by random variables selected from probability distribution. So decision making methods under uncertainty are needed.
To deal with this uncertainty and imprecise data, the concept of fuzziness has been applied. This impreciseness has been handled
so far by using the concept of fuzzy mathematics. In fuzzy approach the imprecise parameters are replaced by fuzzy sets with
known membership function or by fuzzy numbers. But it is very difficult to construct a suitable membership function for the
imprecise parameters. So consideration of interval number of the parameter in the proposed model is more realistic. In this paper
we present parameters as an interval number in parametric function form and then solve this parametric problem by geometric
programming (GP) technique. Geometric programming is an efficient technique for solving particular type non linear
optimization problems. Since late 1960’s, Geometric Programming (GP) used in various field (like OR, Engineering science etc.).
The theory of Geometric Programming (GP) first emerged in 1961 by Duffin and Zener and it further developed by Duffin. Duffin
R J, Peterson E L, Zener C M studied Geometric Programming-Theory and Application[4]. It has certain advantages over the
other optimization methods. The advantage is that this method converts a problem with highly non-linear and inequality

constraints (primal problem) to an equivalent problem with linear and equality constraints (dual problem). It is easier to deal with
the dual problem consisting linear and equality constraints than the primal problem with non-linear and inequality constraints.
Kotchenberger [8] was first used GP method to solve the basic inventory problem. Warral and Hall [18] utilized this technique to
solve a multi-item inventory problem with several constraints. This method is now widely used to solve the optimization problem
in inventories. Friedman (1978) developed continuous time inventory model with time varying demand. Ritchie (1984) studied in
inventory model with linear increasing demand[12-13]. D.Datta and Pravin Kumar published several papers of fuzzy inventory
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Asim Kumar Das et al, Journal of Global Research in Mathematical Archives, 16-21

with or without shortage[5]. S. Islam, T.K. Roy (2006) presented a fuzzy EPQ model with flexibility and reliability consideration
and demand depended unit Production cost under a space constraint[9-10]. Geometric programming plays an important role for
optimization for all above models developed by the authors.
The rest of the paper is organized as follows: In section II, we introduce some basic concepts and definition of interval number.
Imprecise EOQ model using interval number is discussed in section III. In section IV, numerical example has been presented for
different values of the parameter p  (0, 1) to illustrate the proposed model. Finally, conclusion and future research are drawn in
Section V.
II. Some basic concept and definition

Pre-requisite mathematics
In this section we discuss some preliminary mathematics which we have used to study the imprecise EOQ model.
Definition 1 (Interval number): An interval number A is represented by closed interval [
and defined by A=[
=
 R}, where R is the set of real numbers and
{x:
are the left and right limit of the interval number
respectively.
Now we define interval-valued function which will be used to present an interval number.
Definition 2 (Interval-valued function): Let c, d >0 and consider the interval is of the form [c, d], the interval-valued function of
the interval is represented as h (p) =
for p  [0, 1].
Now we present some arithmetic operations on interval valued functions. Let A=[
so that
> 0.
Addition: A+B = [
(p) =

+[

=[

where

Subtraction:

[

number A  B is given by h (p) =
Scalar multiplication:

βA = β[

number βA is given by h (p) =

 al

and

 au .

be two interval number

. The interval-valued function for the interval number A +B is given by h

and

A  B =[

and B= [

.
=[

. Provided

, where
={

[
[

if β ≥ 0 and

and

> 0. The interval-valued function for the interval
.

provided

> 0. The interval-valued function for the interval

h (p) =

if β < 0 where

and

,

III. IMPRECISE ECONOMIC ORDER QUANTITY (EOQ) OR ECONOMIC LOT SIZE (ELS) MODEL
Economic Order Quantity (EOQ) or Economic Lot Size (ELS) model with uniform rate of demand infinite production rate and has
no shortages.
We derive an imprecise EOQ formula and the minimum total average cost under the following assumption and notation:
1) The inventory system involves only one item.
2) The demand rate is known constant and occurs uniformly but imprecise, is presented in terms of interval number.
3) Holding cost per unit quantity per unit time is known constant but imprecise, is presented in terms of interval number.
4) Fixed ordering cost or set up cost per order is known constant but imprecise is presented in terms of interval number.
5) Production or the re-supply of the item is instantaneous (i.e. production or re-supply rate is infinite.)
6) Lead time is zero.
7) Shortages are not allowed.
Notations:

D̂  [

Ĉ1  [

: Demand rate, units per unit time.

: Constant carrying cost or holding cost per unit quantity per unit time,

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Asim Kumar Das et al, Journal of Global Research in Mathematical Archives, 16-21

Ĉ3  [

: Fixed ordering cost or set up cost per order ( or production run)

Q : Order quantity (or lot size),i.e., number of units ordered per order (units)
U : Purchasing cost per unit quantity.
q(t) : Inventory level at any time, t≥0.
T: Cycle of length of the given inventory.
TAC(T): Total average cost per unit time.
The parametric functional form of the interval numbers
,

D(p) =

=

,

D̂ , Ĉ1 , Ĉ3 are presented as follows
=

for

p [0,1]

For p= 0 and 1 we have the left end and right end value of the interval.
Now q(t) is the inventory level at time
over [
is

[

, the differential equation for the instantaneous inventory level q(t) at any time t

D̂ for

(3.1)

With initial condition q(0) = Q

(3.2)

And boundary condition q(T) = 0

(3.3)

From (4.1) , using initial condition (3.2) , we get

D̂

(3.4)

Using boundary condition (3.3) in (3.4), we get Q =

Ĉ1 ∫ (

Ĉ1 ∫

Holding Cost (HC)

Ĉ1 D̂

[ using (3.5) ]

D̂ ฀T

(3.5)

D̂ )

(3.6)

Total cost = Set up cost + Holding cost + Purchasing cost

Ĉ1 D̂

Ĉ3

(3.7)

Total Average cost i.e. TAC(T)

Ĉ3

[ Ĉ3

]

Ĉ1 D̂
Ĉ1 D̂

D̂

[using(3.5)]

(3.8)

So, Problem is

D̂
Such that

Ĉ3

Ĉ1 D̂

(3.9)

.

(3.9) be an unconstrained non-linear programming problem. Geometric Programming (GP) technique has been applied to solve
the non-linear programming problem (3.9). From (3.9), ignoring constant term we take
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Asim Kumar Das et al, Journal of Global Research in Mathematical Archives, 16-21

Ĉ3
Such that

Ĉ1 D̂

(3.10)

.

Here TAC(T) = U D̂ ฀ + TAC1(T).

(3.10) can be taken as a primal geometric programming problem with degree of difficulty (DD)
Degree of Difficulty (DD)
Its dual geometric programming problem is

Ĉ3

(

Ĉ1 D̂

)

;

are dual variables

Such that

( Normality condition)

For primal variable T,

( Orthogonality condition)
(Positivity condition).

Solving,

we get

(

2Cˆ 3

Ĉ1 D̂

)

√ Ĉ 1

D̂

D̂

D̂

= U D1 

1 p

D̂



D2  p 

2 C1L C3L D1

 C
1 p
2

U
1

C3U D2



p
2

√ Ĉ 1 Ĉ3

D̂
(3.11)

Again from primal dual relations, we get

Ĉ3
Ĉ1 D̂
√

Ĉ3
Ĉ1 D̂
√

And

  C  C

2 C3L

=

Ĉ3 D̂

1 p
2

U
3



2 C3L D1

=

Ĉ1

p
2

L
1

 C
1 p
2

D1

U
3

 C

D2

p 1
2

U
1

D2





p
2

(3.12)

 C  C 
p

p 1
2

L
1

p
U 2
1

[using (3.5)] .

(3.13)

The representation of T*, Q* and TAC*(T) in terms of interval-valued function is as follows
√
√

=

  C  C

2 C3L



1 p
2

2 C3L D1

=

= U D1 

U
3

 C

√

1 p

p
2

1 p
2

D2  p 

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U
3



L
1

D2

D1

 C
p 1
2

U
1





p
2

, for p  [0, 1].

(3.14)

p
U 2
1

, for p  [0, 1].

(3.15)

C3U D2



D2

 C  C 
p

2 C1L C3L D1

L
1

p 1
2

 C
1 p
2

U
1

p
2

, for p  [0, 1].

(3.16)
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Asim Kumar Das et al, Journal of Global Research in Mathematical Archives, 16-21

IV. NUMERICAL EXAMPLE:
A manufacturing company produce an item whose demand is almost 50-60 units per year. The production cost of one item is $
200 and the holding cost per item is near about $ (10-15) per year. The replacement is instantaneous and no shortages are allowed.
We shall now calculate for different values of p  ฀(0,1)

1) The economic lot size,( Q)

2) Optimal total average cost (TAC)
3) Optimal time period (T)
Where the set up cost is assumed almost $(100-110).
For p= 0 and 1 we have the left end and right end value of the interval, which is as our classical problem, here we study only for
intermediate values of in between 0 and 1.
This is given in terms of tables representation and graphical representation.

TAC * (T ) for different values of ‘p’

TABLE: Optimum value of T*, Q* and
D(p)=
p

L 1 p

(D )

0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9

U

(D )

C1 ( p)

p

50.92
51.8569
52.811
53.7827
54.7723
55.78
56.8063
57.8516
58.916

=

C3 ( p )

=

(C1L )1 p (C1U ) p

(C3L )1 p (C3U ) p

10.4138
10.8447
11.2935
11.7608
12.2474
12.7542
13.282
13.8316
14.404

100.958
101.924
102.901
103.886
104.881
105.885
106.899
107.923
108.957

rough sketch of p versus T* graph

Q*

0.617073
0.602062
0.587419
0.573131
0.559192
0.545590
0.532319
0.519371
0.506739

31.4214
31.2211
31.0222
30.8245
30.6282
30.4330
30.2391
30.0465
29.8550

0.60

Q*

T*

0.58

31.6

12400

31.4

12200

31.2

12000

31.0

11800

30.8

11600

TAC*(T)

0.62

30.6

0.56
30.4

0.54

10511.2
10710
10912.5
11119.1
11329.6
11544.1
11762.9
11985.9
12213.2

rough sketch of p versus Q* graph

rough sketch of p versus Q* graph

0.64

TAC * (T )

T*

11400
11200

30.2

11000

30.0

10800

29.8

10600

0.52

0.50

10400

29.6

0.0

0.2

0.4

0.6

0.8

1.0

0.0

0.2

0.4

values of p

Fig: 1

0.6

0.8

1.0

0.0

Fig:2

From the above table and figure it is clear that

0.2

0.4

0.6

0.8

1.0

values of p

values of p

Fig:3

T , Q decreases with increasing of ‘p’, while TAC * (T ) increases with
*

*

increasing of ‘p’ (as expected).
V. Conclusion:
In this paper, a single item imprecise economic order quantity model is developed in more realistic sense. Here the cost
components like holding cost per item, set up cost are considered here as an imprecise parameter. In real life situation the demand
rate of an item may not be precise at any time, which has some flexibility due some reason i.e it may be considered as an
© JGRMA 2016, All Rights Reserved

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Asim Kumar Das et al, Journal of Global Research in Mathematical Archives, 16-21

imprecise constant in realistic sense. This impreciseness are represented here with an interval number which are introduced in
terms of parametric functional form of an interval. This new approach of handling the impreciseness has an advantage that when p
varies from 0 to 1, we get lower bound, intermediate value and upper bound of the interval valued number to express the nature of
the model. The objective goals are not always precise. The authority allows some flexibility to attain his target. Using this
procedure an authority can achieve their target by varying the level of optimistic value of p from 0 and 1. The model is illustrated
with a practical example. Geometric Programming (GP) method is used here to solve the problem. The model can be easily
extended to any other inventory problems with other constraints. The method presented here is quite general and can be applied to
the real life inventory problems faced by the practitioners in industry or in other areas.

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